# Please, help me solve this differential equation

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1. Oct 27, 2015

### Remixex

1. The problem statement, all variables and given/known data
OK, this differential equation was technically created by me, because i need to clear my doubts.
Y'' + sqrt(X)*Y' + X^3*Y=3sin(x)
and actually just any initial conditions as long as the solution is something i can understand, let me expand my doubt further.

I've never solved a second order ODE with functions as coefficients, i've always done it with constant coefficients (because those are the ones that describe oscillations), i usually solve with variation of parameters, Y=Yh+Yp where Yh=C1*Y1(x) +C2 Y2(x) calculated with the "D" operator, and then Yp=U1*Y1 + U2*Y2 and U1 and U2 are calculated integrating the division of the respective wronskians.
The only ways i have to solve these "variable coefficients" equations are via Cauchy-Euler methods (X^m) or Riccati equation knowing one solution.
I actually had to be told this is solved via variation of parameters, and i have no idea how.

2. Relevant equations
Y'' +q(x)Y' + p(x) Y = g(x)
(In fact if you have any other example stored in that clarifies my doubt i'd greatly appreciate it, as long as it fits this equation above)

3. The attempt at a solution
The differential operator would be D^2 + sqrt(x)*D +X^3=0, but really i don't know how to solve for "D" (to get the general expression for Y1 and Y2) if i also have to solve for X
Thanks in advance, this might actually be a very basic doubt (it has been in my mind a long while, i know how to solve some PDEs and membrane problems with different boundary conditions but not this)

2. Oct 28, 2015

### HallsofIvy

Staff Emeritus
I would look for power series solution to that.

3. Oct 28, 2015

### Remixex

Oh snap, it seems i just created an equation that can't be solved teh way i want to x.x, thanks for your help, i'll go with my doubt directly to the teacher